A078870 -id:A078870 - cp's OEIS frontend

A078868 Decimal concatenations of the quadruples (d1,d2,d3,d4) with elements in {2,4,6} for which there exists a prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4).

2424, 2462, 2466, 2642, 2646, 2664, 2666, 4242, 4246, 4264, 4624, 4626, 4662, 4666, 6246, 6264, 6266, 6424, 6426, 6462, 6466, 6626, 6642, 6646, 6662, 6664

Offset: 1

Labos Elemer, Dec 19 2002

			4624 corresponds to the quadruple (4,6,2,4). It is in the sequence because the 5 consecutive primes 1597, 1601, 1607, 1609 and 1613 have differences (4,6,2,4).

The least primes corresponding to the quadruples are in A078866. The same primes, in increasing order, are in A078867. The sequences of primes corresponding to the 26 difference patterns are in A022006 (for 2424), A022007 (for 4242) and A078946-A078970. The similarly defined quintuples are in A078870. Cf. A001223.

With[{k = 4}, FromDigits /@ Select[Tuples[Range[2, 6, 2], k], Function[m, Count[Range[k, 10^k], n_ /; Times @@ Boole@ Map[PrimeQ, Prime@ n + Accumulate@ m] == 1] > 0]]] (* Michael De Vlieger, Mar 25 2017 *) (* or *)
FromDigits /@ Union@ Select[ Partition[ Differences@ Prime@ Range[3, 2000], 4, 1], Max@ # <= 6 &] (* Giovanni Resta, Mar 25 2017 *)

Edited by Dean Hickerson, Dec 20 2002

A078872 The quintuples (d1,d2,d3,d4,d5) with elements in {2,4,6} are listed in lexicographic order; for each quintuple, this sequence lists the smallest prime p >= 7 such that the differences between the 6 consecutive primes starting with p are (d1,d2,d3,d4,d5), if such a prime exists.

Original entry on oeis.org

11, 17, 41, 29, 59, 5849, 6959, 599, 149, 3299, 7, 13, 37, 67, 1597, 19, 4639, 43, 17467, 1601, 23, 2333, 593, 6353, 1861, 31, 61, 90001, 32353, 157, 14731, 47, 587, 2671, 3307, 151, 251, 3301

Offset: 1

Labos Elemer, Dec 20 2002

Comment from N. J. A. Sloane, Dec 04 2015: (Start)
The definition of A078872 is fairly subtle.
Step 1: The 3^5 = 243 quintuples (d1,d2,d3,d4,d5) with elements in {2,4,6} are listed in lexicographic order.
Step 2: Study each quintuple in turn.  Look for  the smallest prime p >= 7 such that the differences between the 6 consecutive primes starting with p are (d1,d2,d3,d4,d5).  If there is no such prime move on to next quintuple. If there is at least one such prime, take the smallest one, add it to the sequence, and move on to the next quintuple.
Each quintuple is considered just once, so there are at most 243 terms (in fact there are only 38).
(End)
The 38 quintuples for which p exists are listed, in decimal form, in A078870.

			The term 67 corresponds to the quintuple (4,2,6,4,6): 67, 71, 73, 79, 83 and 89 are consecutive primes.

The quintuples are in A078870. The same primes, in increasing order, are in A078873. The analogous sequences for quadruples and 6-tuples are in A078866 and A078874. Cf. A001223.

Edited by Dean Hickerson, Dec 21 2002

A078869 Number of n-tuples with elements in {2,4,6} which can occur as the differences between n+1 consecutive primes > n+1. (Values of a(11), ..., a(18) are conjectured to be correct, but are only known to be upper bounds.)

Original entry on oeis.org

3, 7, 15, 26, 38, 48, 67, 92, 105, 108, 109, 118, 130, 128, 112, 80, 36, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Dec 19 2002

nonn

The ">n+1" rules out n-tuples like (2,2), which only occurs for the primes 3, 5, 7. All terms from a(19) on equal 0.

An n-tuple (a_1,a_2,...,a_n) is counted iff the partial sums 0, a_1, a_1+a_2, ..., a_1+...+a_n do not contain a complete residue system (mod p) for any prime p.

Eric Weisstein's World of Mathematics, k-Tuple Conjecture

The 26 4-tuples and 38 5-tuples are in A078868 and A078870. Cf. A001359, A008407, A029710, A031924, A022004-A022007, A078852, A078858, A078946-A078969, A020497.

test[tuple_] := Module[{r, sums, i, j}, r=Length[tuple]; sums=Prepend[tuple.Table[If[j>=i, 1, 0], {i, 1, r}, {j, 1, r}], 0]; For[i=1, Prime[i]<=r+1, i++, If[Length[Union[Mod[sums, Prime[i]]]]==Prime[i], Return[False]]]; True]; tuples[0]={{}}; tuples[n_] := tuples[n]=Select[Flatten[Outer[Append, tuples[n-1], {2, 4, 6}, 1], 1], test]; a[n_] := Length[tuples[n]]

Edited by Dean Hickerson, Dec 20 2002

A078871 Decimal concatenations of the 6-tuples (d1,d2,d3,d4,d5,d6) with elements in {2,4,6} for which there exists a prime p >= 7 such that the differences between the 7 consecutive primes starting with p are (d1,d2,d3,d4,d5,d6).

Original entry on oeis.org

242462, 246264, 246266, 246626, 264242, 264246, 264264, 264626, 264666, 266424, 266466, 266646, 424626, 424662, 462462, 462642, 462646, 462664, 462666, 466264, 466266, 466626, 624626, 626424, 626426, 626462, 626466, 626642, 626646, 626664, 642462, 642466, 642646, 646246, 646264, 646266, 646626, 646662, 662642, 662646, 662664, 662666, 664246, 664626, 664662, 666264, 666266, 666462

Offset: 1

Labos Elemer, Dec 20 2002

			For 424662, the first 2 primes with the given differences are 37 and 25767877. For 646626, the least start prime is 6268957.

The least primes corresponding to the 6-tuples are in A078874. The same primes, in increasing order, are in A078875. The similarly defined quadruples and quintuples are in A078868 and A078870. Cf. A001223, A078869.

Edited by Dean Hickerson, Dec 21 2002

A078873 Sorted version of A078872.

Original entry on oeis.org

7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 59, 61, 67, 149, 151, 157, 251, 587, 593, 599, 1597, 1601, 1861, 2333, 2671, 3299, 3301, 3307, 4639, 5849, 6353, 6959, 14731, 17467, 32353, 90001

Offset: 1

Labos Elemer, Dec 20 2002

Each term is the smallest prime p >= 7 such that the differences between the 6 consecutive primes starting with p are (d1,d2,d3,d4,d5), for some quintuple (d1,d2,d3,d4,d5) with elements in {2,4,6}.

			The term 90001 corresponds to the quadruple (6,4,6,2,4): 90001, 90007, 90011, 90017, 90019, 90023 are consecutive primes.

The quintuples are in A078870. The same primes, in lexicographic order of the quintuples, are in A078872. The analogous sequences for quadruples and 6-tuples are in A078867 and A078875. Cf. A001223.

Edited by Dean Hickerson, Dec 21 2002

cp's OEIS Frontend

A078868 Decimal concatenations of the quadruples (d1,d2,d3,d4) with elements in {2,4,6} for which there exists a prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4).

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A078872 The quintuples (d1,d2,d3,d4,d5) with elements in {2,4,6} are listed in lexicographic order; for each quintuple, this sequence lists the smallest prime p >= 7 such that the differences between the 6 consecutive primes starting with p are (d1,d2,d3,d4,d5), if such a prime exists.

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A078869 Number of n-tuples with elements in {2,4,6} which can occur as the differences between n+1 consecutive primes > n+1. (Values of a(11), ..., a(18) are conjectured to be correct, but are only known to be upper bounds.)

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A078871 Decimal concatenations of the 6-tuples (d1,d2,d3,d4,d5,d6) with elements in {2,4,6} for which there exists a prime p >= 7 such that the differences between the 7 consecutive primes starting with p are (d1,d2,d3,d4,d5,d6).

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A078873 Sorted version of A078872.

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